Given an integer n, your task is to count how many strings of length n can be formed under the following rules:

• Each character is a lower case vowel ('a''e''i''o''u')
• Each vowel 'a' may only be followed by an 'e'.
• Each vowel 'e' may only be followed by an 'a' or an 'i'.
• Each vowel 'i' may not be followed by another 'i'.
• Each vowel 'o' may only be followed by an 'i' or a 'u'.
• Each vowel 'u' may only be followed by an 'a'.

Since the answer may be too large, return it modulo 10^9 + 7.

Example 1:

Input: n = 1
Output: 5
Explanation: All possible strings are: "a", "e", "i" , "o" and "u".


Example 2:

Input: n = 2
Output: 10
Explanation: All possible strings are: "ae", "ea", "ei", "ia", "ie", "io", "iu", "oi", "ou" and "ua".


Example 3:

Input: n = 5
Output: 68

Constraints:

• 1 <= n <= 2 * 10^4

## Solution: DP

dp[i][c] := number of strings of length i ends with letter c
ans = sum(dp[n])

Time complexity: O(n)
Space complexity: O(n) -> O(1)

## Solution 2: Matrix multiplication

Time complexity: O(logn)
Space complexity: O(1)

## C++

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